How do non-Trinitarian Christians understand the description of Deity in Isaiah 46:9?

As a nontrinitarian who believes, just as Jesus said, that the Father is "the only true God" (John 17:3), I see no problem at all with this verse. As a student of Biblical Hebrew and Greek, I note that "Elohim" is not a name in Hebrew, just as "king" is not a name in English. Elohim simply means "God" in this context, and the fact that the word used with it here is "me" shows plainly that God is singular.

Many Old Testament passages teach the same truth. The most famous of these is the first part of the Shema.

Hear, O Israel: The LORD our God is one LORD: (Deuteronomy 6:4, KJV)

But the first commandment of the Ten declares the same truth.

Thou shalt have no other gods before me. (Exodus 20:3, KJV)
Thou shalt have none other gods before me. (Deuteronomy 5:7, KJV)

Note also the pronoun God uses in speaking to Moses.

And God said unto Moses, I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you. (Exodus 3:14, KJV)

While some may discuss which tense that verb "I AM" should be translated to in English, whether "am" or "was" or "will be", no one will dispute the fact that it is in first-person common singular form. No Hebrew scholar would say it should be "we are." It is not plural.

In the song of Moses, the rhetorical question is asked:

Who is like unto thee, O LORD, among the gods? who is like thee, glorious in holiness, fearful in praises, doing wonders? (Exodus 15:11, KJV)

The answer, of course, is "no one." There is none who can equal God.

Regarding the verse containing both the form "el" and "elohim," this happens more frequently in poetic passages, and even with God's name (Yahweh/Yah)--see, for example, Psalm 68 which uses, in Hebrew, "Yah", "Yahweh", "Adonai", "Elohim", "El", etc. for variety (not necessarily all in the same verse). It appears to be simply a variation to avoid sounding overly redundant, and some writers used this variation more than others. While they are different words, "el" and "elohim", like "Yah" and "Yahweh", are basically two forms of the same word, akin to "math" versus "mathematics" in English. The longer form, of course, is the more formal and usual in Hebrew.

The Bible all speaks in agreement to the fact that there is only one God, and no one is His equal. Jesus himself declares that...

...my Father is greater than I. (John 14:28, KJV)

Not even Jesus was the equal of God.

. . . for there is one God; and there is none other but he: (Mark 12:32, KJV)

From the perspective of Swedenborgian Christians, who see God as one both in person and in essence, in whom is the Father, the Son, and the Holy Spirit, Isaiah 46:9 simply says what we believe: There is one and only one God.

Emanuel Swedenborg (1688–1772), our great theologian, devotes many pages, even whole chapters, to demonstrating that Jesus Christ, whom he simply calls "the Lord" was and is none other that God himself come to earth. Not the incarnation of some supposed eternal "God the Son," or some supposed second Person of the Godhead come to earth, so that he would represent only one-third of God, but the totality of God come to earth, just as Paul says:

For in him dwells all the fullness of the Godhead bodily (Colossians 2:9)

In a chapter in his book Doctrine of the Lord titled "The Lord Is God Himself, the Origin of and Subject of the Word" (§§37–44 of that book), Swedenborg quotes many passages to demonstrate that the same things that are attributed to the one God in the Old Testament are also attributed to Jesus in the New Testament. For example, in the Old Testament

I am Jehovah, and there is no Savior other than me. (Isaiah 43:11)

Am I not Jehovah? And there is no God other than me; and there is no Savior other than me. Look to me so that you may be saved, all you ends of the earth. (Isaiah 45:21–22)

I am Jehovah your God; there is no Savior other than me. (Hosea 13:4)

Compare those and others like them to these passages in the New Testament:

To you is born this day in the city of David a Savior, who is the Christ, the Lord. (Luke 2:11)

They said to the woman, "It is no longer because of what you said that we believe, for we have heard for ourselves, and we know that this is truly the Savior of the world." (John 4:42)

But it has now been revealed through the appearing of our Savior Jesus Christ, who abolished death and brought life and immortality to light through the gospel. (2 Timothy 1:10)

Again, in the Old Testament:

Thus says the Lord, the King of Israel, and his Redeemer, the Lord of hosts: I am the first, and I am the last; besides me there is no god. (Isaiah 44:6)

Listen to me, O Jacob, and Israel, whom I called: I am he; I am the first, and I am the last. (Isaiah 48:12)

And in the New Testament, as part of John's vision of the risen and glorified Jesus Christ:

When I saw him, I fell at his feet as though dead. But he placed his right hand on me, saying, "Do not be afraid; I am the first and the last." (Revelation 1:17)

Many more examples could be given of the New Testament referring to Jesus in the very same way that the Old Testament refers to God. Put this together with the many passages in both the Old Testament and the New Testament saying that there is only one God, and from a non-trinitarian standpoint that accepts Jesus as the incarnation of God, the only possible conclusion is that Jesus Christ is God himself come to earth.

This is why, after the Resurrection, Thomas addressed Jesus as:

My Lord and my God. (John 20:28)

In summary, once again, Isaiah 46:9 simply says exactly what we believe. No "interpretation" is necessary. Just read the words and believe them.

About the different names for God in the Bible, each of them has its own meaning relating to different attributes of God. But they all refer to the one God, who encompasses all attributes.